A golden rectangle has a length of 10 cm. The ratio of its length to its width is (1 + sqrt 5)/2 what is the width of the rectangle in simplified radical form? I need to know the formula to solve other problems like this.
10/w = (1+√5)/2
w = 10*2/(1+√5) = 20(√5-1)/(5-1) = 5(√5-1) = 10 * (√5-1)/2
How did the 5 - 1 get on the denominator?
On step 2
To find the width of the golden rectangle in simplified radical form, we can use the formula for the ratio of a golden rectangle. The ratio of length (L) to width (W) of a golden rectangle is given by:
L/W = (1 + √5)/2
In this case, the length (L) is given as 10 cm. We need to solve for the width (W).
Let's substitute the given values into the equation:
10/W = (1 + √5)/2
To isolate W, multiply both sides of the equation by W:
10 = (1 + √5)/2 * W
Next, multiply (1 + √5)/2 with W:
10 = (W + √5W) / 2
Multiply both sides of the equation by 2 to get rid of the denominator:
20 = W + √5W
Now, isolate the term with the radical (√5W):
√5W = 20 - W
To get rid of the radical, square both sides of the equation:
5W^2 = (20 - W)^2
Expand the right side of the equation:
5W^2 = 400 - 40W + W^2
Combine like terms:
4W^2 + 40W - 400 = 0
Now, we have a quadratic equation. We can use the quadratic formula to solve for W:
W = (-b ± √(b^2 - 4ac)) / (2a)
Plug in the values for a, b, and c:
W = (-40 ± √(40^2 - 4*4*(-400))) / (2*4)
Simplify:
W = (-40 ± √(1600 + 6400)) / 8
W = (-40 ± √8000) / 8
W = (-40 ± 89.44) / 8
Now, we have two possible widths:
W1 = (-40 + 89.44) / 8 ≈ 6.18 cm
W2 = (-40 - 89.44) / 8 ≈ -14.43 cm
Since the width of a rectangle cannot be negative, the width of the golden rectangle in simplified radical form is approximately 6.18 cm.
The formula used to solve this problem is the quadratic formula: W = (-b ± √(b^2 - 4ac)) / (2a). By substituting the given values into the formula and simplifying the equation, we can find the value of W.